Amenability and paradoxical decompositions for pseudogroups and for discrete metric spaces

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CECCHERINI, Tullio, GRIGORCHUK, Rostislav, DE LA HARPE, Pierre

CECCHERINI, Tullio, GRIGORCHUK, Rostislav, DE LA HARPE, Pierre. Amenability and paradoxical decompositions for pseudogroups and for discrete metric spaces. Proceedings of the Steklov Institute of Mathematics, 1999, vol. 224, p. 68-111

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http://archive-ouverte.unige.ch/unige:12574

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FOR PSEUDOGROUPS AND FOR DISCRETE METRIC SPACES

Tullio Ceccherini-Silberstein,

Rostislav Grigorchuk and Pierre de la Harpe

Abstract. This is an expostion of various aspects of amenability and paradoxical decompositions for groups, group actions and metric spaces. First, we review the formalism of pseudogroups, which is well adapted to stating the alternative of Tarski, according to which a pseudogroup without invariant mean gives rise to paradoxical decompositions, and to dening a Flner condition. Using a Hall-Rado Theorem on matchings in graphs, we show then for pseudogroups that existence of an invariant mean is equivalent to the Flner condition; in the case of the pseudogroup of bounded perturbations of the identity on a discrete metric space, these conditions are moreover equivalent to the negation of the Gromov's so-called doubling condition, to isoperimetric conditions, to Kesten's spectral condition for related simple random walks, and to various other conditions. We dene also the minimal Tarski number of paradoxical decompositions associated to a non-amenable group action (an integer4), and we indicate numerical estimates (Sections II.4 and IV.2). The nal chapter explores for metric spaces the notion of superamenability, due for groups to Rosenblatt.

T.C.-S.: Dipartimento di Matematica Pura ed Applicata,

Universita degli Studi dell' Aquila, Via Vetoio, I-67100 L'Aquila, Italy E-mail : [email protected]

R.G.: Steklov Mathematical Institute, Gubkina Str. 8, Moscow 117 966, Russia.

E-mail : [email protected] and [email protected]

P.H.: Section de Mathematiques, C.P. 240, CH-1211 Geneve 24, Suisse.

E-mail : [email protected]

I. Introduction

The present exposition shows various aspects of amenability and non-amenability. Our initial motivation comes from a note on the Banach-Tarski paradox where Deuber, Simonovitz and Sos indicate one kind of paradoxical decomposition for metric spaces, in relation with what they call an \exponential growth"property [DeSS]. Our rst purpose is to revisit their work which, in our view, relates paradoxical decompositions with amenabilityrather than with growth (see in particular Observation 33 below).

For this, we recall in Chapter II the formalism of set-theoretical pseudogroups which is well adapted to showing the many aspects of amenability: existence of invariant nitely additive measures, absence of paradoxical decomposition, existence of Flner sets and isoperimetric estimates. We also state one version of the basic Tarski alternative : a pseudogroup is either amenable or paradoxical.

In Chapter III, we specialize the discussion to metric spaces and pseudogroups of bounded perturbations of the identity; metric spaces, there, are discrete (except at the very end of the chapter). On one hand, this is an interesting class, with many examples given by nitely generated groups. On the other hand, it provides a convenient setting for proving Flner characterization as stated in Chapter II. We discuss also the Kesten characterization in terms of simple random walks.

For a groupGwhich is not amenable, we estimate in Chapter IV the Tarski numberT(G)2 f4;5;:::;1g; an integer which indicates the minimal number of pieces involved in a paradoxical decomposition of G: It is known that ^T(G) = 4 if and only if G has a subgroup which is free non abelian. We show that one

The authors acknowledge support from the \Fonds National Suisse de la Recherche Scientique".

Typeset by^A^M^S-TEX

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has 5 T(G) 34 [respectively 6 T(G) 34] for some torsion-free groups [resp. for some torsion groups] constructed by Ol'shanskii [Ol1], and 6 T^?B(m;n)14 forB(m;n) a Burnside group onm2 generators of odd exponentn665 [Ady].

Building upon the seminal 1929 paper by von Neumann [NeuJ], Rosenblatt has dened for groups a notion of superamenability. He has shown that superamenable groups include those of subexponential growth, and it is not known whether there are others. In Chapter V, we investigate superamenability for pseudogroups and for discrete metric spaces; in particular, we describe a simple example of a graph which is both superamenable and of superexponential growth.

We are grateful to Joseph Dodziuk, Vadim Kaimanovich, Alain Valette and Wolfgang Woess for useful discussions and bibliographical informations, as well as to Laurent Bartholdi for Presentation 12, Example 74 and his critical reading of a preliminary version of this work.

II. Amenable pseudogroups

II.1. Pseudogroups

1. Denition.

In the present set-theoretical context, a pseudogroup ^G of transformations of a set X is a set of bijections:S^!T between subsetsS;T ofX which satises the following conditions (as listed, e.g., in [HS1]):

(i) the identityX^!X is in^G;

(ii) if :S^!T is in^G;so is the inverse^?1:T ^!S;

(iii) if:S^!T and:T ^!U are in ^G; so is their composition:S ^!U;

(iv) if:S!T is inG and ifS⁰ is a subset ofS;the restrictionjS⁰ :S⁰!(S⁰) is inG; (v) if:S!T is a bijection between two subsetsS;T ofX and

if there exists a nite partitionS =t¹jnSj withjSj in Gforj2 f1;:::;ng; then is in^G (where^tdenotes a disjoint union).

Property (v) expresses the fact that ^G is closed with respect to nite gluing up; together with (iv), they express the fact that, for a bijection;being in^Gis in some sense a local condition.

For:S^!T in^G;we write also() for the domainSofand!() for its rangeT:For \a pseudogroup

G of transformations of a setX", we write shortly \a pseudogroup (^G;X)", or even \a pseudogroup^G".

2. Examples.

(i) Any action of a group Gon a set X generates a pseudogroup ^GG;X: More precisely, a bijection :S ^!T is in ^GG;X if there exists a nite partitionS =^t¹jnSj and elementsg¹;:::;gn ²G such that (x) = gj(x) for all x ² Sj;j ^{2 f}1;:::;n^g: If there exists such a ; the subsets S;T of X are sometimes said to beG-equidecomposable (or \endlich zerlegungsgleich"in [NeuJ]).

In caseG=X acts on itself by left multiplications, we write^GG instead of^GG;G:

(ii) Piecewise isometries of a metric spaceX constitute a pseudogroup^Pi^Is(X);generated (in the obvious way) by the partial isometries between subsets ofX:Observe that it may be much larger than the pseudogroup associated as in the previous example with the group of isometries ofX; see for example the metric space obtained from the real line by gluing two hairs of dierent length at two distinct points of the line.

(iii) For a metric space X; the pseudogroup^W(X) of bounded perturbations of the identity consists of bijections:S ^!T such that sup_x²_Sd((x);x)<¹:This is the main example in [DeSS], where it is called the group of wobbling bijections; the notion seems to come from the important work by Laczkovich [Lacz].

3. Remarks.

The above notion of pseudogroup of transformations is strongly motivated by the study of Banach-Tarski paradoxes, as shown by the rst three observations below.

(i) The very denition of a paradoxical decomposition with respect to a group action involves the associated pseudogroup as in Example 2.i.

(ii) Pseudogroups are easily restricted on subsets as in Example 2.iv. This is important for the study of superamenability (see Chapter V below).

(iii) Pseudogroups are easily induced on oversets, as in Example 2.v. This is useful in the setting of a pseudogroup constituted by bijections with domains and range required to be in a given algebra (or-algebra) of subsets of X (for example the measurable sets of a measure space), and in corresponding variations on the Tarski alternative [HS1].

(iv) For a pseudogroup (^G;X);the set

R = (x;y)²XXthere exists^{2 G} such thatx²() andy=(x)

is an equivalence relation. A natural problem is to study the existence of measuresonX such that, for each measurable subset AofX of measure zero, the saturated set ^fx2Aj there existsa2Awith (x;a)^{2 R g} has also measure zero, see [CoFW], [Kai2], [Kai3].

(v) In a topological context, Conditions (iv) and (v) in Denition 1 are usually replaced by a condition involving restrictions to open subsets; see [Sac] and page 1 of [KoNo].

(vi) Consider a metric spaceX;the pseudogroup^W(X) of Example 2.iii, and a subspaceAofX:It is then remarkable (though straightforward to check) that the pseudogroup^W(A) coincides with the restriction of

W(X) toA in the sense of Example 2.iv.

II.2. Amenability and paradoxical decompositions - the Tarski alternative Let (^G;X) be a pseudogroup. We denote by ^P(X) the set of all subsets ofX:

4. Denitions.

^A^G-invariant meanonX is a mapping:^P(X)^![0;1] which is (fa) nitely additive: (S¹^[S²) = (S¹) +(S²) forS¹;S²X withS¹^\S²=^;; (in) invariant: ^?!() = ^?()for all^{2 G};

(no) normalised: (X) = 1:

More generally, for A X;a ^G-invariant mean on X normalised on A is a mapping : ^P(X) ^! [0;¹] which satises Conditions (fa) and (in) above, as well as

(no⁰) (A) = 1:

The pseudogroup G is amenable if there exists a G-invariant mean on X; and the triple (G;X;A) is amenableif there exists aG-invariant mean onX normalised onA: These notions are essentially due to von Neumann [NeuJ].

5. Denition.

A paradoxical^G-decomposition ofX is a partitionX =X¹tX²such that there exist_j2 G

with(_j) =X_j and!(_j) =X (j= 1;2):

A pseudogroup (^G;X) is paradoxical if it has a paradoxical^G-decomosition, or equivalently (because of Theorem 7 below) if it is not amenable.

6. Remarks.

(i) There cannot exist such paradoxical^G-decomposition if ^G is amenable.

This is obvious, because (with the notations of Denitions 4 and 5) one cannot have 1 = (X) = (X¹) +(X²) = 2 !

It is remarkable that there is no further obstruction, as Theorem 7 shows.

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(ii) LetG;Hbe two pseudogroups of transformations of the same setX;with G H: IfHis amenable, then so is G; if G is paradoxical, then so isH: This will be used for example in the proof of Theorem 25 (Item 36).

(iii) In short-hand, Denition 5 reads 2[X]= [^!! X]:It has variations in the literature; for example, one may ask (n+ 1)[X]^!! n[X], or more precisely :

there exists an integern1 and elements¹;:::;N^{2 G} such that

j fj^{2 f}1;:::;N^{g j}x²(j)^{g j}n+ 1 for allx²X;namely^P^k_j⁼¹[(j)](n+ 1)[X];and

j fj2 f1;:::;Ng jx2!(j)g j nfor allx2X;namely^P^k_j⁼¹[!(j)]n[X]:

Then Remark (i) still holds for the same obvious kind of reason. Indeed, the variation is equivalent to Denition 5, as can be seen either with manipulations a la Cantor-Bernstein (see for example [HS1]) or as a consequence of the following theorem.

7. Theorem (Tarski alternative).

Let^G be a pseudogroup of transformations of a setX:Exactly one of the following holds :

- either^G is amenable,

- or there exists a paradoxical^G-decomposition ofX:

Let moreoverAbe a non-empty subset of X and let^G⁽A⁾ be the pseudogroup obtained by restriction of^G; as in Example 2.iv. Exactly one of the following holds :

- either there exists a^G-invariant mean onX normalised onA;

- or there exists a paradoxical^G⁽_A⁾-decomposition ofA:

The theorem originates in Tarski's work : see [Tar3], as well as earlier papers by Tarski ([Tar1], [Tar2]).

One proof for pseudogroups has been written up in [HS1]. Its starting point is an application of the Hahn-Banach theorem, to the Banach space`¹(X) of bounded real-valued functions onX;to the subspace d¹(X) of nite linear combinations of functions of the form ^?!()?^?() for some 2 G (where (A) denotes the characteristic function of A), and to the open cone C of functions F 2 `¹(X) such that infx²XF(x)>0; one has to observe thatG has an invariant mean if and only ifd¹(X)\ C =;:This proof uses also ideas of Banach, Cantor-Bernstein, Hausdor, Konig, Kuratowski and von Neumann.

We give here another proof, based on what we call the Hall-Rado theorem (Theorem 35), which is essentially the \Konig theorem"of [Wag]. More precisely, the rst statement of Theorem 7 is a straightforward consequence of Theorems 25 and 32, and the second statement follows (see the sketch below).

Much more complete information on all this can be found in Wagon's book (see [Wag], in particular Corollary 9.2 on page 128). Important more recent work in this area include [DouF].

Let us sketch the proof of the second statement of the theorem. Assume that the pseudogroup^G⁽A⁾is not paradoxical, so that, by the rst statement, there exists a ^G⁽A⁾-invariant mean A : ^P(A)^! [0;1]: Dene then a mapping:^P(X)^![0;¹] as follows; for a subset Y ofX;if there exists a partitionY =^t¹jnYj

and elements ¹ :Y¹ ! B¹;:::;_n : Y_n ! B_n in ^G withB¹;:::;B_n A; then set (Y) =^Pⁿ_j⁼¹_A(B_j);

otherwise, set (Y) = ¹: Then one checks that is well dened and that it is a ^G-invariant mean on X normalised onA:

8. Remark.

A famous theorem of E. Hopf can be expressed very much like Tarski's alternative.

LetT :X ?!X be an ergodic non-singular transformation of a nite probability space (X;B;m);with m non-atomic. Let [[T]] denote the set of all 1-1 non-singular transformations : U !V such that (x) belongs to the T-orbit ofx for allx 2U (withU;V 2 B); this [[T]] is the full groupoid of T of Katznelson and Weiss [KaWe, page 324]. For two measurable subsets A;B of X;say that A is dominated by B; and writeAB;if there exists a measurable subsetB⁰ ofB withm(BnB⁰)>0 and a bijective transformation :A?!B⁰ in [[T]].

Hopf alternative.

(i) In the situation above, exactly one of the following holds : - there exists aT-invariant probability measure on (X;^B) equivalent tom;

- one hasX X:

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(ii) Also, exactly one of the following holds :

- there exists aT-invariant innite measure on (X;^B) equivalent tom;

- one hasX X;and there exists A^{2 B} withm(A)>0 such thatAis not dominated byA:

In other words, (i) says that there is a nite invariant measure in the measure classmif and only ifX itself is not \Hopf-compressible", and (ii) that there is an innite invariant measure in the measure classmif and only ifXis Hopf-compressible and some measurable subset ofXof positive measure is not Hopf-compressible [Weis].

If there exists aT-invariant probability measure [respectively innite measure] on (X;^B) equivalent tom;

thenT is said to be of type II¹ [resp. of TypeII¹].

II.3. The case of groups

For any groupG;we consider rst the pseudogroup^GG which is associated with the action ofGon itself on the left, as in Example 2.i.

Let nowGbe a group generated by a nite setS:Let`_S :G!^N denote the corresponding word length function; thus `_S associates to g2Gthe smallest integern0 for which there exist s¹;:::;s_n 2S[S^?1 withg=s¹:::sn:LetdLand dR denote respectively the left and right invariant metrics onGdened by

dL(x;y) = `S^?x^?1y dR(x;y) = `S^?xy^?1 for allx;y²G:

Besides^GG;we consider also the pseudogroup^Pi^Is(G) of piecewise isometries of the metric space (G;dL), as in Example 2.ii, as well as the pseudogroup^W(G) of bounded perturbations of the identity of the metric space (G;dR);as in Example 2.iii. It is easy to check that the pseudogroup^W(G) does not depend on the choice ofS:

9. Observation.

With the notations above, one has^GG=^W(G) for any nitely generated groupG:

Proof. It is obvious that^GG^W(G):Conversely, let:U !V be in^W(G): Set k = sup_x

2Ud((x);x) B = fg2Gj`S(g)kg

and observe thatB is a nite subset ofG:For eachg2B;set Ug = ^fx²U^j(x) =gx^g: One hasU =^tg²BUg and(x) =gx for allx²Ug:Hence ^{2 G}G:

It is clear that ^GG ^Pi^Is(G): It is also clear that ^GG ⁶= ^Pi^Is(G) in general (example : for G = ^Z generated by^f1^g;the isometryn^{7! ?}nis not in^G^Z).

10. Denition.

A groupGis amenable if the pseudogroup^GG is amenable.

If G is nitely generated, the previous observation shows that one may equivalently dene G to be amenable if the pseudogroupW(G) is amenable.

11. On the class of amenable groups.

Amenability may be viewed as a niteness condition. One of the main problems is to understand various classes of amenable groups, for example those which are nitely generated or nitely presented. (Recall that a group is amenable if and only if all its nitely generated subgroups are amenable; see Theorem 1.2.7 in [Gre1] and Observation 19 below.)

The following question, implicit in [NeuJ], was formulated explicitely by Day, at the end of^x4 in [Day1] : does every non-amenable group contain a free group on 2 generators ? As much as we know and despite

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several misleading allusions in the literature to some \von Neumann conjecture", von Neumann himself has never conjecturedthat a non-amenable group should contain a non-abelian free subgroup !

Day's question was answered negatively by A. Yu. Ol'shanskii [Ol1], Adyan [Ady2] and Gromov [Gro2, Corollary 5.6.D]; the rst two use cogrowth criteria (see Item 52 below) and Gromov uses Property (T). For innite groups, this Property (T) of Kazhdan [Kaz] is (among other things) a strong form of non-amenability : see [Sch] and [CoWe]. However, when restricted to the class of linear groups (i.e. of groups which have faithful nite-dimensional linear representations), Day's question can be answered positively : this follows from an important result due to Tits [Tit].

M. Day has dened the class EG of \elementary amenable groups", which is the smallest class of groups which contains nite groups and abelian groups, and which is closed under the four operations of (i) taking subgroups, (ii) forming factor groups, (iii) group extensions and (iv) upwards directed unions. He has asked (again in [Day1]) whether the class EG coincides with the class AG of all amenable groups (see also [Cho]).

Today, we know that there are nitely generated groups in AG which are not in EG; this has rst been shown using growth estimates ([Gri2], [Gri3]), and more recently by an elegant argument of Stepin (see [Ste], based on [Gri2]).

One knows also nitely presented groups in AG which are not in EG; more precisely, the nite presentation G =

*

a;b;c;d;t a²=b²=c²=d²=bcd= (ad)⁴= (adacac)⁴= 1 t^?1at=aca t^?1bt=d t^?1ct=b t^?1dt=c

+

denes an amenable group which is not elementary amenable ([Gri6], [Gri7]).

12. Bartholdi's presentation.

It has later been shown that the groupGof [Gri6] has a presentation with two generators only (namely aand t) and four relations of total length 109 = 2 + 19 + 32 + 56: Here are Bartholdi's computations, whereT stands fort^?1:

The relationsc = aTata; d= tcT and b =Tctshow rst that the relations c² =d² = b² = 1 may be deleted in the presentation above, and second that the generatorsb;c;dmay also be deleted. Thus

G =

*

a;t a²=TctctcT = (atcT)⁴= (atcTacac)⁴= 1 T²ct²=tcT

+

where c holds foraTata: The relation TctctcT = 1 impliesT²ctctc = 1 =tcT²ctc(by conjugation), hence also (usingc^?1 =c)

1 = ^?T²ctctc^?tcT²ctc^?1 = T²ct²(tcT)^?1

using free simplications, so that the relation T²ct² =tcT may also be deleted. Finally, one observes that atcT is conjugate toTatc= (Tata)²so that (atcT)⁴= 1 may be written (Tata)⁸= 1;and one observes also that atcTacacis equal toataTataTaaTataaaTata;so is conjugate toT²ataTat²aTata:One obtains nally Bartholdi's presentation

G = a;t a²=TaTatataTatataTataT = (Tata)⁸= (T²ataTat²aTata)⁴= 1:

13. Categorical considerations.

For a given integerk;letFkbe the free group onkgenerators^fs¹;:::;sk^g

and let Xk denote the space of all marked groups on k generators, namely of all data Fk ?, where indicates a homomorphism onto. There is an appropriate topology onXk;for which two quotients:Fk ? and⁰:F_k ?⁰are \near"each other if the corresponding Cayley graphs have balls of \large"radius around the unit element which are isomorphic. This makes X_k a compact space; one shows for example that the closure of the subset ofX_kcorresponding to nite groups contains the subset ofX_kcorresponding to residually nite nitely presented groups. For details, see [Gri2], [Cha] and [Ste].

It would be interesting to nd pairs (Y;Z) where

Y is a compact subspace ofXk;

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Z is a \small"(e.g. countable) subset ofY;consisting of amenable groups,

Y nZ consists of non-elementary amenable groups, or more generally the set of elementary amenable groups inY nZ is of rst category.

The point is that the spaceY contains a denseG consisting of amenable groups which are not elementary amenable. (As usual aG inY is a countable intersection of open subsets of Y:)

One such pair has been constructed in [Gri2] and analized in [Ste], with Z a countable set of virtually 2-step solvable groups and withY nZ consisting of innite torsion groups. Understanding other such pairs would probably help us understanding the closures of AGk and of EGk in Xk; where AGk [respectively EGk] denotes the subspace ofXk containing marked groups:Fk ? with ? amenable [resp. elementary amenable].

14. Variation on one question of Day.

Let us denote by BG the smallest class of groups containing nitely generated groups of subexponential growth (see Denition 64) and closed with respect to the four operations of Day listed in 11 above, namely with respect to (i) taking subgroups, (ii) forming factor groups, (iii) group extensions and (iv) upwards directed unions.

Question: does one have BG=AG ?

15. Other denitions of amenability for groups; topological groups.

The natural setting for amenability of groups is that of topological groups, mainly locally compact groups. A substancial part of the theory consists in showing the equivalence of a large number of denitions.

Let G be a Hausdor topological group. Denote by ^C^b(G) the Banach space of bounded continuous functions fromGto^C; with the supremum norm. For^{2 C}^b(G) andg²G;letg ^{2 C}^b(G) be the function x^!f(g^?1x): Denote by^UC^b(G) the closed subspace of^C^b(G) of functions for which the mappingg^7!g from Gto ^C^b(G) is continuous. The following are known to be equivalent (see Theorem 3 in [Day2] and Theorem 4.2 in [Ric2]) :

there exists a left-invariant mean on^UC^b(G);

any continuous actionGQ^!QofGby ane transformations of a non-empty compact convex subsetQof a Hausdor locally convex topological vector space has a xed point.

The groupGis amenable if these properties hold. In caseGis assumed to be locally compact, here is a short list of other equivalent properties :

there exists a left-invariant mean on^C^b(G);

there exists a left-invariant mean onL¹(G);

the unit representation of Gis weakly contained in the left regular representation ofGonL²(G);

for any continuous actionGX ^!X ofGby homeomorphisms of a non-empty compact spaceX;

there exists aG-invariant probability measure onX:

The last point, onG-invariant measures, goes back to a paper by Bogolyubov, see [Bogl], quoted by Anosov [Ano]. This paper, published in Ukrainian in 1939, has remained unnoticed; the paper does not quote von Neumann [NeuJ], and it is conceivable that Bogolyubov has introduced independently the notion of amenability. About relations between amenability, growth and existence of invariant measures, we would also like to quote [Bekl].

The list above is very far from being complete ! (See 16; other items could be : several formulations of the Flner property for locally compact groups, the Reiter-Glicksberg property, the existence of approximate units in the Fourier algebra, ... .) See, e.g., the books [Gre1], [Pat] and [Wag], as well as [Rei, Chapter 8], [Eym2], [Zim, Chapter 4], [Wag, in particular Theorem 10.11] and [Lub, Chapter 2]. In case of a countable group (with the discrete topology), here is the most recent characterization of amenability with which one of the authors has been involved : a countable groupG is amenable if and only if, for any action ofGby homeomorphisms on the Cantor discontinuumK;there exists a probability measure onKwhich is invariant byG[GiH2].

We would like to point out that some attention has been given to topological groups which are not locally compact (in [Ric2, ^x4] among other places). For example, let Û(^H)st be the group of unitary operators on a separable innite dimensional Hilbert space ^H; with the strong topology; then Û(^H)st is amenable, namely there exists a left invariant mean on ÛC^b(Û(^H)st); but there does not exist any left invariant

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mean onCb(U(H)st):Moreover, this group does have closed subgroups which are not amenable; indeed, if

H=`²(Fn) for a free groupFnof rankn2;thenU(H)sthas clearly a discrete subgroup isomorphic toFn; as observed in [Har3]. Here is another example involving non locally compact topologies; letGbe the group of real points of an^R-algebraic group and let ? be a subgroup ofGwhich is dense for the Zariski topology;

if ? is amenable, so isG(see [Moo], and Theorem 4.1.15 in [Zim]).

Let us mention the following : for a locally compact groupGwhich is almost connected (this means that the quotient ofGby the connected component of 1 is compact), the three properties

Gis amenable,

Gdoes not contain a discrete subgroup which is free on 2 generators, G=r(G) is compact,

are equivalent. This is due to Rickert : Theorem 5.5 in [Ric2], building on [Ric1]; see also Theorem 3.8 in [Pat]. Recall that the solvable radical r(G) of a locally compact group Gis the largest connected closed normal solvable subgroup of G [Iwa]. (One may dene similarly the amenable radical of G as the largest amenable closed normal subgroup ofG; see Lemma 1 of^x4 in [Day1] and Proposition 4.1.12 in [Zim].)

This result of Rickert reduces in some sense the problem of understanding the class of amenable locally compact groups to totally disconnected groups; we believe moreover that the most important (and dicult) part of the problem is that which concerns nitely generated groups.

16. Cohomological denitions of amenability.

There are various (co)homological characterizations of amenability.

One is that of Johnson : a groupGis amenable if and only ifH¹(`¹(G);M) is reduced to^f0^gwhenever M is aG-module dual to some BanachG-moduleM [Joh]. It follows that the bounded cohomology of an amenable group is always reduced to^f0^g; this is given by Gromov (Section 3.0 in [Gro1]) together with a reference to an unpublished explanation of Philip Trauber - hence the name \Trauber theorem".

Another one is in terms of \uniformly nite homology"; it applies to nitely generated groups, and indeed to metric spaces in a much broader class. Such a spaceX is not amenable if and only if the groupH⁰^uf(X) is reduced to ^f0^g(in this statement, one may take^R as coecients, or equivalently^Z); this is one way to express that Flner condition does not hold inX [BlW1].

It seems also appropriate to quote here a theorem of Brooks : let Gbe the covering group of a normal coveringM of a compact manifoldX; thenGis amenable if and only if 0 is in the spectrum of the Laplace- Beltrami operator acting on the space of square-integrable functions on M (see [Bro], or the exposition in [Lot]).

There are other conditions in terms of other \coarse"(co)homology theories of the groups, or in terms of K-theory of appropriate algebras associated to the group (see various preprints by G. Elek, including [Ele2]).

Let us mention that there are interesting cohomological consequences of amenability. For example, letG be a group which has an Eilenberg-MacLane spaceK(G;1) which is a nite complex; ifGis amenable, then Ghas Euler characteristic(G) = 0 (a particular case of Corollary 0.6 of Cheeger and Gromov [ChGr], who use`²-cohomology methods, and also a result of B. Eckmann, who uses other methods [Eck]). Also, let G be the fundamental group of some closed 4-manifoldM; ifGis innite and amenable, then(M)0 [Eck].

17. Variations on amenability of groups.

There are standard variations on the pseudogroup^GG and the notion of amenability.

One is to consider the pseudogroup ^GGG associated as in Example 2.i with the action of GG onG dened by (x;y)g=xgy^?1: It is classical that^GGG is amenable if and only if^GG is amenable. In other words : Ghas a left invariant mean if and only ifGhas a two-sided invariant mean (Lemmas 1.1.1 and 1.1.3 in [Gre1]).

Another variation is to consider the action ofGonG^nf1^gdened byxg=xgx^?1and the notion of inner amenabilityfor a group. It is obvious that an amenable group is inner amenable. Straightforward examples (such as non-trivial direct products of free groups and amenable groups) show that there are non-amenable groups which are inner amenable. More on this in [BeHa], [E], [GiH1] and [HS2].

A third variation is to consider a subgroupHofGand the pseudogroup^GG=H associated with the natural action of G on G=H: The subgroupH is said to be co-amenable in G if ^GG=H is amenable. There is a comprehensive analysis of this notion in [Eym1]; see also [Bekk], in particular Theorem 2.3. In caseG=Fm

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is a free group of nite rank, a criterion for co-amenability of a subgroup in terms of cogrowth is given in [Gri1]

(see Item 52 below). One may generalize actions of GonG=H to actions ofG on locally compact spaces;

co-amenability ofH is then a particular case of a notion of amenability for actions known as amenability in the sense of Greenleaf [Gre2].

The notion of amenability for a group and that of co-amenability for a subgroup may both be viewed as particular cases of a notion for G-mappings, for which we refer to [AnaR]. In case of a group Gwith the discrete topology, it can be dened as follows. LetX;Y be two Borel spaces given with measure classes; and with actions of Gby non-singular invertible Borel mappings, and let:X ^!Y be a surjective Borel mapping such that () = ; thus there is a canonical linear isometric mapping by which we identify the Banach space L¹(Y;) to a closed G-invariant subspace of L¹(X;): Say the mapping is amenable if there exists a G-equivariant linear mapping E : L¹(X;)^!L¹(Y;) which is a conditional expectation, namely which is positive and which restricts to the indentity onL¹(Y;): Example 1 : X = Gand Y is reduced to one point; thenX ^! Y is amenable if and only if G is amenable. Example 2 : X =G=H for a subgroupH ofGand Y is reduced to a point; then X !Y is amenable if and only if H is co-amenable inG: Example 3 : X =GZ for a G-spaceZ (withGacting from the left on itself and diagonally on the productGZ); then the projectionGZ ^!Z is amenable if and only if the action ofGonZ is amenable in the sense of Zimmer [Zim, Section 4.3].

There are other notions, including the three following ones : K-amenability [Cun], weak amenability a la Cowling-Haagerup [CowH], and a-T-menability a la Gromov. (See 7.A and 7.E in [Gro3], and [BekCV]; in fact Gromov rediscovered the class of groups having \Property 3B" of Akemann and Walter in [AkWa].)

II.4. Tarski number of paradoxical group actions

Consider more generally the pseudogroupGG;X associated with a group action GX !X (see again Example 2.i).

18. Denition.

^For^:^S^!^T ⁱⁿ^GG;X;dene the Tarski number ofas the smallest \number of pieces"n such that there exists a partition S = ^t¹jnSj and elements g¹;:::;gn in G with (x) = gj(x) for all x²Sj;j^{2 f}1;:::;n^g:

The Tarski number of a paradoxical^GG;X-decomposition

X = X¹^GX² ; ¹:X¹^!X ; ²:X²^!X

as above is the sum of the Tarski number of¹ and of that of ²: It is clear that such a sum is an integer

4:

When ^GG;X is not amenable, we dene the Tarski number ^T(G;X) of the action GX ^! X as the minimum of the Tarski numbers of the paradoxical^GG;X-decompositions ofX; when^GG;X is amenable, we set^T(G;X) =¹:For a groupGacting on itself by left multiplication, we write^T(G) rather than^T(G;G):

19. Observation.

Let Gbe a group given together with a subgroup G⁰ and a quotient group G^{0 0}: It is straightforward that one has

T(G) ^T(G⁰)

T(G) ^T(G⁰⁰):

For example, for the rst of these inequalities, viewGas a disjoint union of cosets ofG⁰:

Each groupGhas a nitely generated subgroupG⁰ such thatT(G⁰) =T(G): Indeed, assumingGto be non-amenable, consider a paradoxical decomposition

G = X¹^t:::^tXm^tY¹:::^tYn = g¹X¹^t:::^tgmXm = h¹Y¹^t:::^thnYn

containingm+n=T(G) pieces (whereX¹;:::;Xm;Y¹;:::;Yn are subsets ofGand g¹;:::;gm;h¹;:::;hn

are elements ofG). LetG⁰ be the subgroup ofGgenerated by^fg¹;:::;gm;h¹;:::;hn^g:SetX_i⁰ =Xi^\G⁰ for alli^{2 f}1;:::;m^gandY_j⁰=Yj^\G⁰ for allj^{2 f}1;:::;n^g:Then

G⁰ = X¹⁰^t:::^tX_m⁰ ^tY¹⁰:::^tY_n⁰ = g¹X¹⁰ ^t:::^tgmX_m⁰ = h¹Y¹⁰^t:::^thnY_n⁰

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so thatT(G⁰) T(G):With the rst inequality of the present observation, this shows thatT(G⁰) =T(G): (One may observe a fortiori thatX¹⁰;:::;Y_n⁰ are non-empty.) It follows that one has

T(G) = inf^?^T(G⁰)

where the inmum is taken over all nitely generated subgroupsG⁰ ofG:

It should be interesting to study how the Tarski number behaves with respect to other group theoretical constructions such as extensions and HNN-constructions. In particular, for the latter, we have in mind some presentations of the Richard Thompson's F group [CaFP]; recall that F is a group which does not have non-abelian free subgroups, which is a HNN-extension of itself [BrGe], thatF is inner-amenable [Jol], that F has non-abelian free subsemigroups so that it is not supermanenable (see Chapter V below), and that one does not know whetherF is amenable or not.

20. Proposition (Jonsson, Dekker).

For a group G; one has ^T(G) = 4 if and only if G contains a non-abelian free subgroup.

Proof. For the free groupF² on 2 generatorsg andh;it is classical that ^T(F²) = 4; see, e.g., Figure 4.1 in [Wag]. We recall this as follows. Set

A¹ = W^?g A² = W^?g^?1

B¹ = W^?h^[1;h^?1;h^?2;:::

B² = W^?h^?1nh^?1;h^?2;:::

whereW(x) denotes the subset ofF²consisting of reduced words on^fg;hgwithx as rst letter on the left, forx2 fg;g^?1;h;h^?1g:Then

F² = A¹^GA²^GB¹^GB² = A¹^GgA² = B¹^GhB²: It follows that^T(F²) = 4:

Observation 19 shows that^T(G) = 4 for any groupGcontaining a subgroup isomorphic toF².

Conversely, let G be a group with ^T(G) = 4; so that there exist subsets X¹;X²;Y¹;Y² and elements g¹;g²;h¹;h²in Gsuch that

G = X¹^GX²^GY¹^GY² = g¹X¹^Gg²X² = h¹Y¹^Gh²Y²: Setg=g¹^?1g² andh=h^?1¹ h²: Then, one has successively

X¹ = GⁿgX² = gX¹^GgY¹^GgY²

X¹ gX¹ ::: g^k^?1X¹ g^kYj (k1 and j= 1;2) X² = Gng^?1X¹ = g^?1X²^Gg^?1Y¹^Gg^?1Y²

X² g^?1X² ::: g^?^k⁺¹X² g^?^kYj (k1 and j= 1;2) so that

g^kYj X¹^[X² for all k²^Z; k⁶= 0 and j= 1;2: One has similarly

h^kXj Y¹^[Y² for all k²^Z; k⁶= 0 and j= 1;2:

Hence g and hgenerate in Ga free subgroup of rank 2; by a classical lemma going back essentially to F.

Klein, and sometimes known as the \table-tennis lemma"(see, e.g., [Har4]).

The argument above is our rephrasing of the proof of Theorem 4.8 in [Wag].

Proposition 20 is an unpublished work from the 40's by B. Jonsson (a student of Tarski) and is a particular case of results of Dekker published in the 50's. For precise references, see the Notes of Chapter 4 in [Wag].

Let us also mention that, for a groupGcontaining a non abelian free group and for an actionGX^!X with stabilizers^fg²G^jgx=x^g which are abelian for allx ²X;the corresponding Tarski number is also given by^T(G;X) = 4 (Theorem 4.5 in [Wag]).

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21. Proposition.

For a non-amenable torsion groupG;one hasT(G)6:

Proof. By Proposition 20 we know that^T(G)5:We assume that ^T(G) = 5;and we will reach a contra- diction.

The hypothesis implies that there exist subsetsX¹;X²;Y¹;Y²;Y³ and elementsg¹;g²;h¹;h²;h³ inGsuch that G = X¹^GX²^GY¹^GY²^GY³ = g¹X¹^Gg²X² = h¹Y¹^Gh²Y²^Gh³Y³:

Letndenote the order ofg⁺g^?1¹ g²:As in the proof of Proposition 11, one has X¹ gX¹ ::: gⁿ^?1X¹ gⁿY¹^GY²^GY³: But nowgⁿ= 1 and this is absurd. Hence^T(G)>5:

22. Question.

Does there exist a groupGwith Tarski number^T(G) equal to 5 ? to 6 ? More generally, what are the possible values of^T(G) ?

II.5. Flner condition for pseudogroups

Let (^G;X) be a pseudogroup of transformations. For a subset^Rof^G and a subsetAofX;we dene the

R-boundary ofAas

@^RA =

(

x2XnA

there exists2 R [ R^?1 such that x2() and (x)²A

)

:

23. Denition.

The pseudogroup (G;X) satises the Flner condition if

for any nite subset^Rof^G and for any real number >0 there exists a nite non-empty subsetF =F(^R;) ofX such that j@^RFj < jFj

where^jF^jdenotes the cardinality of the setF:

24. Ahlfors and Flner.

Ideas underlying the Flner condition go back at least to Ahlfors. (Flner does notrefer to this work.) Ahlfors denes an open Riemann surface S to be regularly exhaustible if, for some complete metricgin the conformal class dened by the complex structure ofS;there exists a nested sequence ¹²:::of domains with smooth boundaries such that^S_n¹n is the whole surface and such that

nlim^!1^j@n^jg jn^jg = 0

where ^j^jg denotes the area of a domain and where^j@^jg denotes the length of its boundary, both with respect tog:(A lemma of Ahlfors shows that this does not depend on the choice ofg:) These sequences may be used to dene averaging processes, as Ahlfors did rst and as Flner did later.

Using this notion, Ahlfors has developped a geometric approach to the Nevanlinna theory of distribution of values of meromorphic functions, known as Ahlfors theory of covering surfaces. In particular, he gave a generalization of the second main theorem of Nevanlinna on defect. (See Section 25 in Chapter III of [Ahl];

see also Chapter XIII in [Nev], Chapter 5 in [Hay], Theorem 6.5 on page 1223 of [Oss], [Sto] and [ZoKe].) Amenability of coverings of Riemann surfaces can also be expressed in terms of Teichmuller spaces [McM2].

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25. Theorem.

A pseudogroup of transformations is amenable if and only if it satisfes the Flner condition.

Flner's original proof (for a group acting on itself by left multiplications) goes back to 1955 [Fol]. The proof has been simplied by Namioka [Nam] (who generalized Flner's result to one-sided cancellative semi- groups), and extended to group actions by Rosenblatt [Ros1]; the best place to read it is probably Section 2.1 of [Co1]. In case of a groupGacting by conjugation onG^nf1^g;the proof can also be found in [BeHa], and it applies verbatim to an action ofGon any setX:All these references use essentially techniques of functional analysis. (See also Wagon's comment about the implication (6) =⁾(1) in Theorem 10.11 of [Wag].)

The proof below, in Items 26 and 36, uses completely dierent techniques.

26. Beginning of the proof of Theorem 25.

We prove here the implication \Flner condition ⁾ existence of an invariant mean".

Let ^M(X) denote the set of all means onX; namely of all nitely additive probability measures onX (see Conditions (fa) and (no) in Denition 4). Let`¹(X) denote the Banach space of all bounded functions onX;with the norm of uniform convergence; it is standard¹ that^M(X) can be identied with a subset of the unit ball in the dual space of`¹(X): It is also standard that the weak-topology makes^M(X) into a compact space.

For each nite non-empty subsetF ofX;we consider the mean _F :

8<

:

P(X) ^?! [0;1]

A ^7?! ^jA^\F^j

jF^j in^M(X):Consider also the set

N = ^f(^R;)^{2 G}^R ^{j R} is nite and >0^g ordered by

(^R;) (^R⁰;⁰) if ^{R R}⁰ and ⁰:

Notations being as in the denition of the Flner condition (which is now assumed to hold),

(*) ^?_F^(R_;⁾(R;^)2N

becomes a net. By compacity of^M(X);this net has a cluster point, say(we use the terminology of [Kel, Chapter 2]). The proof consists in showing thatis^G-invariant; in other words, given a subsetAofX and a transformation in^G withA();one has to show that^?(A)=(A):

We choose a number >0:Asis a cluster point of the family ();there exists (R;)2 N such that (i) (^R;) (^f^g;); i:e:; ^{R 3} and ;

(ii) ^j_F^(R_;⁾(A)^?(A)^j ; (iii) F^(R;⁾^?(A)^?^?(A) : From now on, we writeF instead ofF(R;): Dene

Ai;i = ^fa²A^ja²F and (a)²F^g Ai;o = ^fa²A^ja²F and (a)²@^RF^g Ao;i = fa2Aja2@^RF and (a)2Fg

Ao;o = fa2Aja =2F and (a)2=Fg

1See footnote 37 in [NeuJ], where von Neumann refers in turn to Lebesgue's \Lecons sur l'integration"(1905).

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(think of \inside"for \i"and of \outside"for \o"). Observe thatA=Ai;i^tAi;o^tAo;i^tAo;o;with the rst three sets being nite. Observe also that

(iv) A\F = Ai;i^GAi;o so that jA\Fj = jAi;i^j+jAi;o^j

(v) induces a bijection Ai;i^G^?1(Ao;i)^!(A)^\F so that ^j(A)^\F^j = ^jAi;i^j+^jAo;i^j

(vi) @^RF @^f_;^?1^gF (Ai;o)^[Ao;i

so that ^jAi;o^j+^jAo;i^j 2^j@^RF^j 2^jF^j: It follows from (iv) to (vi) that

(vii) ^j(A)^\Fj ? jA\Fj 2jFj:

Using the denition of the meanF and the conclusion of the Flner condition, one may rewrite (vii) as (viii) F^?(A)?F(A) 2

so that one obtains nally

^?(A)^?(A) 2+ 2 4

using (ii), (iii) and (viii):As the choice ofis arbitrary, this ends the proof of one implication of Theorem 25.

27. Remark.

In case of a locally nite graph X with nitely many orbits of vertices under the full automorphism group (for example in case of a Cayley graph), Flner condition is equivalent to the existence of a nested sequence F¹ F² ::: of nite subsets of the vertex set X⁰ such that ^[n¹F_n = X⁰ and limn^!1^j@Fn^j=^jFn^j= 0; see our Section III.2 for amenable graphs and for the notation@Fn, and Theorem 4.39 in [Soa] for the equivalence.

In the case of a groupGacting on a setX;the Flner condition is most often expressed in a way involving the symmetric dierence between a nite subsetF ofX and its imagegF by someg²G; for the equivalence of this with the analogue of our Denition 23, see Proposition 4.3 in [Ros1].

For groups, Flner condition implies the existence of Flner sets with extra tiling properties, and this is useful for showing extensions to amenable groups of the Rohlin theorem from ergodic theory [OrWe].

III. Amenability and paradoxical decompositions for metric spaces

III.1. Gromov condition and doubling condition LetX be a metric space and letddenote the distance onX:

ForS;TX, a mapping:S^!T (not necessarily a bijection) is a bounded perturbation of the identity if sup_x²_Sd((x);x)<¹:We will denote by

B(X)

the collection of all these maps. (This would be an example of a \pseudo-semi-group", but we will not use this term again below.)

As in Example 2.iii, we denote by^W(X) the pseudogroup of all bijections, between subsets ofX;which are bounded perturbations of the identity.

For a subsetAofX and a real numberk >0;we denote by

Nk(A) = ^fx²X^jd(x;A)k^g thek-neighbourhood ofAinX: